Rearranging Harder Formulae Step-by-Step Examples with Worksheet

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Rearranging Harder Formulae

  • Rearranging harder formulae is a valuable skill that empowers us to manipulate equations, isolate variables, and uncover deeper insights into mathematical relationships.
  • This skill has practical applications across diverse fields, enabling us to solve complex problems and make informed decisions based on mathematical principles.

In this article, we will discuss: 

  1. What is Rearranging Harder Formulae?
  2. Steps for Rearranging the Formula to Change the Subject of the Formula

Here is one more link to practice a few extra questions: Maths Genie Rearranging Harder Formulae Questions

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What is Rearranging Harder Formulae?

  • Rearranging Harder Formulae refers to the process of manipulating a given formula to make a specific variable the subject of the formula.
  • This process is used extensively in mathematics and science to solve problems where it is necessary to change the variable in a formula to obtain the desired result.

Steps for Rearranging the Formula to change the Subject of Formula

  • The following steps can be used to rearrange a formula to change the subject of the formula:

Step #1: Identify the variable you want to make the subject of the formula.

Step #2: Write down the given formula and identify the variable you want to make the subject.

Step #3: Isolate the variable you want to make the subject by performing inverse operations on both sides of the equation.

Step #4: Simplify the equation by performing any necessary algebraic manipulations.

Step #5: Check your answer by substituting the new expression for the variable back into the original equation to ensure it is correct.

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Solved Example: 

Question 1: Rearrange the formula v = u + at to make t the subject.

Solution: 

  • Step #1: Identify the variable you want to make the subject of the formula, which is t.
  • Step #2: Write down the given formula and identify the variable you want to make the subject.

v = u + at

  • Step #3: Isolate the variable t by performing inverse operations on both sides of the equation.

v = u + at

v – u = at

(t = (v – u) / a)

  • Step #4: Simplify the equation by performing any necessary algebraic manipulations.

t = (v – u)/a

  • Step #5: Check your answer by substituting the new expression for the variable back into the original equation to ensure it is correct.

v = u + a((v – u)/a)

v = u + v – u

v = v

The solution is correct.

 

Question 2: Rearrange the formula P = (4L + 2W) / 3 to make L the subject.

Solution: 

  • Step #1: Identify the variable you want to make the subject of the formula, which is L.
  • Step #2: Write down the given formula and identify the variable you want to make the subject.

P = (4L + 2W)/3

  • Step #3: Isolate the variable L by performing inverse operations on both sides of the equation.

P = (4L + 2W) / 3

3P = 4L + 2W

(4L = 3P – 2W)

(L = (3P – 2W) / 4)

  • Step #4: Simplify the equation by performing any necessary algebraic manipulations.

L = (3P – 2W) / 4

  • Step #5: Check your answer by substituting the new expression for the variable back into the original equation to ensure it is correct.

P = (4((3P – 2W) / 4) + 2W) / 3

P = (3P – 2W + 2W) / 3

P = P

The solution is correct.

 

Question 3: Make x the subject of the formula

Rearranging Harder Formulae Questions

Solution: 

  • Step #1: Multiply both sides of the equation by (x – b) to eliminate the denominator on the right side.

a(x – b) = x + c

  • Step #2: Expand the left side of the equation.

ax – ab = x + c

  • Step #3: Move all the terms containing x to the left side of the equation and all the constant terms to the right side.

ax – x = ab + c

  • Step #4: Factor out x on the left side of the equation.

x(a – 1) = ab + c

  • Step #5: Divide both sides of the equation by (a – 1) to isolate x.

x = (ab + c) / (a – 1)

Therefore, x = (ab + c) / (a – 1) is the formula for x in terms of a, b, and c.

Conclusion

  • Rearranging harder formulae is a valuable skill that empowers us to manipulate equations, isolate variables, and uncover deeper insights into mathematical relationships.
  • By mastering the art of rearranging harder formulae, we develop critical thinking and problem-solving abilities that are essential for success in mathematics and scientific disciplines.
  • The ability to rearrange equations allows us to simplify complex expressions, facilitating a clearer understanding of the underlying concepts and principles.
  • Through the mastery of rearranging harder formulae, we unlock the potential to tackle challenging mathematical problems, optimize processes, and make accurate predictions based on mathematical relationships.

Practice Questions: Rearranging Harder Formulae

Question 1: Rearrange the formula v = πr2h to solve for h.

Question 2: Rearrange the formula A = lw + 2lh + 2wh to solve for w.

Question 3: Rearrange the formula T = 2π√(L/g) to solve for L.

Question 4: Rearrange the formula E = mc2 + mv2/2 to solve for v.

Question 5: Rearrange the formula P = 2πh + 2πr to solve for h.

Question 6: Rearrange the formula v = πr2h + (4/3)πr3 to solve for r.

Question 7: Rearrange the formula A = 2πrh + 2πr2 to solve for h.

Question 8: Rearrange the formula F = G(m1m2/r2) to solve for r.

Question 9: Rearrange the formula v = (4/3)πr3 – (4/3)πr13 to solve for r.

Question 10: Rearrange the formula A = (s/2)(a + b) to solve for b.

Solutions:

Question 1: Rearrange the formula v = πr2h to solve for h.

Solution:

Step #1: Start with the formula

v = πr2h

Step #2: Divide both sides by πr2

v / (πr2) = h

The rearranged formula is:

h = v / (πr2)

Question 2: Rearrange the formula A = lw + 2lh + 2wh to solve for w.

Solution:

Step #1: Begin with the formula

A = lw + 2lh + 2wh

Step #2: Subtract lw and 2lh from both sides.

A – lw – 2lh = 2wh

Step #3:Factor out w on the right side.

A – lw – 2lh = w(2h)

Step #4: Divide both sides by (2h).

(A – lw – 2lh) / (2h) = w

The rearranged formula is:

w = (A – lw – 2lh) / (2h)

Question 3: Rearrange the formula T = 2π√(L/g) to solve for L.

Solution:

Step #1:Start with the formula

T = 2π√(L/g)

Step #2:Divide both sides by 2π.

T / (2π) = √(L/g)

Step #3: Square both sides of the equation.

(T / (2π))2 = L/g

Step #4: Multiply both sides by g.

g(T / (2π))2 = L

The rearranged formula is:

L = g(T / (2π))2

Question 4: Rearrange the formula E = mc2 + mv2/2 to solve for v.

Solution:

Step #1: Begin with the formula

E = mc2 + mv2/2

Step #2:Subtract mc2 from both sides

E – mc2 = mv2/2

Step #3:Multiply both sides by 2/m.

(2/m)(E – mc2 ) = v2

Step #4: Take the square root of both sides.

√[(2/m)(E – mc2)] = v

The rearranged formula is:

v = √[(2/m)(E – mc2)]

Question 5: Rearrange the formula P = 2πh + 2πr to solve for h.

Solution:

Step #1:Start with the formula

P = 2πh + 2πr

Step #2:Subtract 2πr from both sides.

P – 2πr = 2πh

Step #3:Divide both sides by 2π.

(P – 2πr) / (2π) = h

The rearranged formula is:

h = (P – 2πr) / (2π)

Question 6: Rearrange the formula v = πr2h + (4/3)πr3 to solve for r.

Solution:

Step #1:Begin with the formula

v = πr2h + (4/3)πr3

Step #2:Subtract (4/3)πr3 from both sides.

v – (4/3)πr3 = πr2h

Step #3:Divide both sides by πh.

(v – (4/3)πr3) / (πh) = r2

Step #4: Take the square root of both sides.

√[(V – (4/3)πr3) / (πh)] = r

The rearranged formula is:

r = √[(V – (4/3)πr3) / (πh)]

Question 7: Rearrange the formula A = 2πrh + 2πr2 to solve for h.

Solution:

Step #1: Start with the formula

A = 2πrh + 2πr2

Step #2: Subtract 2πr2 from both sides.

A – 2πr2 = 2πrh

Step #3: Divide both sides by 2πr.

(A – 2πr2) / (2πr) = h

Step #4: Simplify the expression on the left side.

A/(2πr) – r = h

The rearranged formula is:

h = A/(2πr) – r

Question 8: Rearrange the formula F = G(m1m2/r2) to solve for r.

Solution:

Step #1: Begin with the formula

F = G(m1m2/r2)

Step #2: Multiply both sides by r2.

Fr2 = G(m1m2)

Step #3: Divide both sides by F.

r2 = G(m1m2)/F

Step #4: Take the square root of both sides.

r = √(G(m1m2)/F)

The rearranged formula is:

r = √(G(m1m2)/F)

Question 9: Rearrange the formula v = (4/3)πr3 – (4/3)πr13 to solve for r.

Solution:

Step #1: Start with the formula

v = (4/3)πr3 – (4/3)πr13

Step #2: Add (4/3)πr13 to both sides.

v + (4/3)πr13 = (4/3)πr3

Step #3: Multiply both sides by 3/(4π).

(3/(4π))(v + (4/3)πr13) = r3

Step #4: Take the cube root of both sides.

r = ∛[(3/(4π))(v + (4/3)πr13)]

The rearranged formula is:

r = ∛[(3/(4π))(v + (4/3)πr13)]

Question 10: Rearrange the formula A = (s/2)(a + b) to solve for b.

Solution:

Step #1: Begin with the formula

A = (s/2)(a + b)

Step #2: Divide both sides by (s/2).

A / (s/2) = a + b

Step #3: Simplify the expression on the left side.

2A / s = a + b

Step #4: Subtract a from both sides.

(2A / s) – a = b

The rearranged formula is:

b = (2A / s) – a